3 added 249 characters in body

A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.

Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. These are in fact all (see Hartshorne IV.3.9).

For this reason, in any characteristic we do not use lines to get embeddings in projective space.

But this actually has consequences. Many classical theorems in algebraic geometry do not use 'transcendental methods', i.e. the only result they use is that the base field is algebraically closed, and so they can be applied in finite characteristic. Or they cannot?

Here is where characteristic $2$ breaks down the standard results. For instance, when embedding blow-ups of $\mathbb{P}^2$ in $\mathbb{P^n}$ we use linear systems of conics and cubics in $\mathbb{P}^2$ to separate the points, but this is not possible in characteristic $2$ (have a look at Beauville IV.4 if you do not know how linear systems can embed spaces). This means that in characteristic 2 we cannot interpret cubic surfaces ni $\mathbb{P}^3$ in terms of blow-ups of the projective plane in 6 points in general position and viceversa.

OK, enough intro. My question is: "Are there other examples of results which do not apply in characteristic 2 due to other reasons not involving embeddings in projective space?"

Also, I am looking for results that do not hold in low characteristic. i.e. I know that vanishing theorems do not hold in characteristic p in general, but I am looking for pathologies for some but not all finite characteristic (usually 2, or 3).

I suspect 'probably yes but not many', since I cannot come up with any, but if it turns out there are lots, maybe I'll make this question community wiki.

2 forgot smoothness

A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.

Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. These are in fact all (see Hartshorne IV.3.9).

For this reason, in any characteristic we do not use lines to get embeddings in projective space.

But this actually has consequences. Many classical theorems in algebraic geometry do not use 'transcendental methods', i.e. the only result they use is that the base field is algebraically closed, and so they can be applied in finite characteristic. Or they cannot?

Here is where characteristic $2$ breaks down the standard results. For instance, when embedding blow-ups of $\mathbb{P}^2$ in $\mathbb{P^n}$ we use linear systems of conics and cubics in $\mathbb{P}^2$ to separate the points, but this is not possible in characteristic $2$ (have a look at Beauville IV.4 if you do not know how linear systems can embed spaces). This means that in characteristic 2 we cannot interpret cubic surfaces ni $\mathbb{P}^3$ in terms of blow-ups of the projective plane in 6 points in general position and viceversa.

OK, enough intro. My question is: "Are there other examples of results which do not apply in characteristic 2 due to other reasons not involving embeddings in projective space?"

I suspect 'probably yes but not many', since I cannot come up with any, but if it turns out there are lots, maybe I'll make this question community wiki.

1

# Which 'well-known' algebraic geometric results do not hold in characteristic 2?

A curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.

Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. These are in fact all (see Hartshorne IV.3.9).

For this reason, in any characteristic we do not use lines to get embeddings in projective space.

But this actually has consequences. Many classical theorems in algebraic geometry do not use 'transcendental methods', i.e. the only result they use is that the base field is algebraically closed, and so they can be applied in finite characteristic. Or they cannot?

Here is where characteristic $2$ breaks down the standard results. For instance, when embedding blow-ups of $\mathbb{P}^2$ in $\mathbb{P^n}$ we use linear systems of conics and cubics in $\mathbb{P}^2$ to separate the points, but this is not possible in characteristic $2$ (have a look at Beauville IV.4 if you do not know how linear systems can embed spaces). This means that in characteristic 2 we cannot interpret cubic surfaces ni $\mathbb{P}^3$ in terms of blow-ups of the projective plane in 6 points in general position and viceversa.

OK, enough intro. My question is: "Are there other examples of results which do not apply in characteristic 2 due to other reasons not involving embeddings in projective space?"

I suspect 'probably yes but not many', since I cannot come up with any, but if it turns out there are lots, maybe I'll make this question community wiki.